This book is devoted to considering in the general case — using typical concrete examples — the motion of machines and mechanisms of impact and vibro-impact action accompanied by a peculiar phenomenon called “impact collapse”. This phenomenon is that after the initial collision, a sequence of repeated gradually quickening collisions of decreasing-to-zero intensity occurs, with the final establishment of protracted contact between the interacting bodies. The initiation conditions of the impact collapse are determined and calculation techniques for the quantitative characteristics of the corresponding motions are proposed. It is shown that impact collapse occurs frequently in reality.The book is intended for experts in applied mathematics, mechanics and mechanical engineering. The resulting formulas and plots are directly usable in engineering practice.